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The sequence a1, a2, a3, ..., an of n integers is such that [#permalink]
 23 Mar 2009, 11:17
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The sequence a_1, a_2, a_3, ... a_n of n integers is such that a_k=k if k is odd, and a_k=-a_{k-1} if k is even. Is the sum of the terms in the sequence positive? (1) n is odd. (2) a_n is positive
Last edited by Bunuel on 29 Dec 2012, 04:29, edited 1 time in total.
Renamed the topic, edited the question and added OA.
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Re: DS GMAT perp sequence [#permalink]
23 Mar 2009, 22:48
My Answer:
D. EACH statement ALONE is sufficient.
Explanation:
As,
ak=k if k is odd
Hence, a1=1
a3=3, a5=5 etc.
And as
ak=-ak-1 if k is even
Hence, a2=-a1=-1
a4=-a3=-3
a6=-a5=-5 etc.
So, if n is even,
a1+a2+....+an = a1+(-a1)+a3+(-a3)+....+an-1+(-an-1) = 0
All terms get canceled.
And if n is odd,
a1+a2+....+an = a1+(-a1)+a3+(-a3)+....+an-2+(-an-2)+an = an
As, ak=k if k is odd, an = n if n is odd
Only an remains at the end, which is a positive number equal to n.
Hence each statement satisfies it individually.
Now tell me the OA pls.
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Re: The sequence a1, a2,a3,....an of n integers is such that [#permalink]
28 Dec 2012, 07:34
Can you explain this statement a little further, I just do not understand how you arrived at that statement Hence, a2=-a1=-1. Regards
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Re: The sequence a1, a2,a3,....an of n integers is such that [#permalink]
29 Dec 2012, 04:30
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KevinBrink wrote: Can you explain this statement a little further, I just do not understand how you arrived at that statement Hence, a2=-a1=-1. Regards Stem says that a_k=k if k is odd. So, for k=1=odd we have that a_1=1. The sequence a_1, a_2, a_3, ... a_n of n integers is such that a_k=k if k is odd, and a_k=-a_{k-1} if k is even. Is the sum of the terms in the sequence positive?We have following sequence: a_1=1; a_2=-a_1=-1; a_3=3; a_4=-a_3=-3; a_5=5; a_6=-a_5=-5; ... Basically we have a sequence of positive and negative odd integers: 1, -1, 3, -3, 5, -5, 7., -7, 9, -9, ... Notice than if the number of terms in the sequence (n) is odd then the sum of the terms will be positive, for example if n=3 then a_1+a_2+a_3=1+(-1)+3=3, but if the number of terms in the sequence (n) is even then the sum of the terms will be zero, for example if n=4 then a_1+a_2+a_3+a_4=1+(-1)+3+(-3)=0. Also notice that odd terms are positive and even terms are negative. (1) n is odd --> as discussed the sum is positive. Sufficient. (2) a_n is positive --> n is odd, so the same as above. Sufficient. Answer: D. Hope it's clear.
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Re: The sequence a1, a2, a3, ..., an of n integers is such that [#permalink]
30 Dec 2012, 05:13
Accountant wrote: The sequence a_1, a_2, a_3, ... a_n of n integers is such that a_k=k if k is odd, and a_k=-a_{k-1} if k is even. Is the sum of the terms in the sequence positive?
(1) n is odd.
(2) a_n is positive In a sequence it always helps to observer a first few terms. Given these definitions : a1=1, a2=-a1 = -1, a3 = 3, a4=-a3 = -3 so it is clear consecutive terms from begining are canceling each other, i.e., 1-1+2-2+3-3 etc Also, the sum is either positive in which case it is equal to the last odd term or it is zero. So knowing either the term is odd or that last term was positive helps us know that sum of the terms are positive D
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Re: The sequence a1, a2, a3, ..., an of n integers is such that [#permalink]
03 Jan 2013, 22:38
a1 + (-a1) + a3 + (-a3) + .......
1. n is odd --> last term is an which is positive every other term cancels out
2. an can be +ve only if n is odd which will be the last term same as above
hence answer is D
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Re: The sequence a1, a2, a3, ..., an of n integers is such that [#permalink]
14 Feb 2013, 05:55
Now this is what i dont understand. They just mention "k" (i guess constant) but k can take any value -1,-3 or +2. SHouldnt the answer be B then because the statement 2 specifically says that an is positive.
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Re: The sequence a1, a2, a3, ..., an of n integers is such that [#permalink]
14 Feb 2013, 06:05
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Re: The sequence a1, a2, a3, ..., an of n integers is such that [#permalink]
14 Feb 2013, 10:17
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maddyboiler wrote: Now this is what i dont understand. They just mention "k" (i guess constant) but k can take any value -1,-3 or +2. SHouldnt the answer be B then because the statement 2 specifically says that an is positive. dont go into complex things. Just visualize the sequence It can be 2 way 1,-1, 3,-3, 5,-5, 7,-7 ending in negative term The sum will be zero in this case 1,-1, 3,-3, 5,-5, 7 ending in positive term The sum will be the last term of sequence we have asked is the sum positive ? ----------> is the sequence as per 2nd case ? ----------> is the a(n) odd ? or is the a(n) positive ? both the statements answer these questions so both are sufficient
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Re: The sequence a1, a2, a3, ..., an of n integers is such that
[#permalink]
14 Feb 2013, 10:17
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